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Definite integralsCalculusCollegeabout 1:06

Integrate x^2sin(x) from -3 to 4: animated lesson

A narrated, step-by-step definite integrals video lesson with captions. Final answer: -4.680076795.

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Transcript

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.5 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int_{-3}^{4} {x}^{2} \sin\left( x\right)\,dx$
2. Steps, part 1
1. $\int_{-3}^{4} {x}^{2} \sin\left( x\right)\,dx$
  Start with the definite integral
2. $\int u\,dv = uv - \int v\,du$
  Integration by parts (tabular method)
3. $\int {x}^{2} \sin\left( x\right)\,dx = \cos\left( x\right) \left(2-{x}^{2}\right)+2x \sin\left( x\right)$
  Combine the alternating products
4. $F(x) = \cos\left( x\right) \left(2-{x}^{2}\right)+2x \sin\left( x\right)$
  Antiderivative
3. Steps, part 2
1. $F(4) - F(-3) = 3.09659073 - 7.776667525 = -4.680076795$
  Apply the fundamental theorem of calculus
4. The answer
$\int_{-3}^{4} {x}^{2} \sin\left( x\right)\,dx = -4.680076795$
∫ from -3 to 4 = -4.680076795
5. Check
- Differentiated the antiderivative and compared it with the integrand ✓
- Adaptive Simpson's rule gives -4.680076795 ✓

Transcript

The problem. Here is the problem: integrate x to the power 2sin(x) from negative 3 to 4. Evaluate the definite integral of x squared sin(x) from x equals negative 3 to x equals 4.

Steps, part 1. Start with the definite integral. Integration by parts (tabular method). Differentiate the polynomial x squared until it reaches 0, integrate sin(x) the same number of times, and combine with alternating signs. Combine the alternating products. Antiderivative.

Steps, part 2. Apply the fundamental theorem of calculus.

The answer. So the answer is negative 4.680076795. A definite integral gives the signed area between the curve and the x-axis: area above the axis counts as positive and area below counts as negative.

Check. Differentiated the antiderivative and compared it with the integrand, which checks out. Adaptive Simpson prime s rule gives negative 4.680076795, which checks out.

About definite integrals

Evaluate definite integrals with the fundamental theorem of calculus.

Integrate x^2sin(x) from -3 to 4: Definite integrals Lesson